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Current Question (ID: 15943)

Question:

$\text{A transverse wave propagating along the } x\text{-axis is represented by:}$ $y(x, t) = 8.0 \sin\left(0.5\pi x - 4\pi t - \frac{\pi}{4}\right), \text{ where } x \text{ is in meters and } t \text{ in seconds. The speed of the wave is:}$

Options:

1. $4\pi \text{ m/s}$
2. $0.5 \text{ m/s}$
3. $\frac{\pi}{4} \text{ m/s}$
4. $8 \text{ m/s}$

Solution:

$\text{Hint: } v = \frac{\omega}{k}$ $\text{Step: Find the speed of the wave.}$ $\text{The given equation is:}$ $y(x,t) = 8.0 \sin\left(0.5\pi x - 4\pi t - \frac{\pi}{4}\right) \ldots (i)$ $\text{The standard wave equation can be written as,}$ $y = a \sin(kx - \omega t + \phi) \ldots (ii)$ $\text{Comparing the Eq. (i) and (ii), we have,}$ $k = 0.5\pi, \omega = 4\pi$ $\text{So, the speed of the transverse wave is,}$ $v = \frac{\omega}{k} = \frac{4\pi}{0.5\pi} = 8 \text{ m/s}$ $\text{Hence, option (4) is the correct answer.}$

Import JSON File

Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}